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Strongly exposed points in bases for the positive cone of ordered Banach spaces and characterizations of l1(Г)

Published online by Cambridge University Press:  20 January 2009

Ioannis A. Polyrakis
Ioannis A. Polyrakis
Affiliation:
Department of Mathematics, National Technical University, Zografon Campus, 157 73, Athens, Greece
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The study of extreme, strongly exposed points of closed, convex and bounded sets in Banach spaces has been developed especially by the interconnection of the Radon–Nikodým property with the geometry of closed, convex and bounded subsets of Banach spaces [5],[2] . In the theory of ordered Banach spaces as well as in the Choquet theory, [4], we are interested in the study of a special type of convex sets, not necessarily bounded, namely the bases for the positive cone. In [7] the geometry (extreme points, dentability) of closed and convex subsets K of a Banach space X with the Radon-Nikodým property is studied and special emphasis has been given to the case where K is a base for acone P of X. In [6, Theorem 1], it is proved that an infinite-dimensional, separable, locally solid lattice Banach space is order-isomorphic to l1 if, and only if, X has the Krein–Milman property and its positive cone has a bounded base.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 2 , June 1986 , pp. 271 - 282
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Bourgain, J. and Talagrand, M., Dans un espace de Banach reticulé solid, la propriété de Radon-Nikodým et celle de Krein-Milman sont équivalentes, Proc. Amer. Math. Soc. 81 (1981), 93–96.Google Scholar
2.Bourgin, R. D., Geometric Aspects of Convex Sets with the Radon-Nikodym Property (Lecture Notes in Mathematics 993).Google Scholar
3.Jameson, G. J. O., Ordered Linear Spaces (Lecture Notes in Mathematics, 141).Google Scholar
4.Kendall, D., Simplexes and vector lattices, J. London Math. Soc. 37 (1962), 365–371.CrossRefGoogle Scholar
5.Phelps, R. R., Dentability and extreme points in Banach spaces, J. FUNCTIONAL ANALYSIS 16 (1974), 78–90.CrossRefGoogle Scholar
6.Polyrakis, I. A., Lattice Banach spaces order-isomorphic to l 1, Math. Proc. Cambridge Phil. Soc. 94 (1983), 519–522.CrossRefGoogle Scholar
7.Polyrakis, I. A., Extreme points of unbounded, closed and convex sets in Banach spaces, Math. Proc. Cambridge Phil. Soc. 95 (1984), 319–323.CrossRefGoogle Scholar
8.Schmidt, G. C., Extensions theorems for linear lattices with positive algebraic basis, Periodica Mathematica Hungarica 6(4), (1975), 295–307.CrossRefGoogle Scholar